Let now \(\lbrace \alpha _1, \dots , \alpha _\texttt {g}, \beta _1,\dots , \beta _\texttt {g}\rbrace \)  be representing cycles for a canonical homology basis for \(M\) 
such that each \(\beta _j\)  intersects \(\alpha _j\)  once from the right to the left and no other intersections occur (see [1]} for details).
Then there are corresponding harmonic differentials \(dU_{\alpha _j}\) , \(dU_{\beta _j}\)  obtained on choosing \(\gamma =\alpha _j,\beta _j\) 
in the above construction, and these constitute a basis of harmonic differentials associated with the chosen homology basis.
Precisely we have, for \(k,j=1,\dots ,\texttt {g}\) ,
\(\oint _{\alpha _k} (-dU_{\beta _j})=\delta _{kj}, \quad \oint _{\beta _k} (-dU_{\beta _j})=0,\) 
\(\oint _{\alpha _k} dU_{\alpha _j}=0, \qquad \oint _{\beta _k} dU_{\alpha _j}=\delta _{kj}.\) 
